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Statement

Concluding remarks, item (v), as printed on p. 526:

"(v) Is it true that

$$\lim_{k\to\infty}\frac{r(C_{2n+1};k)}{r(C_3;k)}\to0\qquad\text{for }n\ge2.$$

It is not even known at present that

$$\frac{\log r(C_{2n+1};k)}{k}=O(1),\qquad n\ge2."$$

The display combines the limit symbol with an arrow, as printed. Here $r(G;k)$ is the least order forcing a monochromatic $G$ in every $k$-coloring (p. 515), so $r(C_3;k)$ is the $k$-color Ramsey number of the triangle and the question is Problem 554 with the site's notation $R_k(C_{2n+1})/R_k(K_3)$. The paper's earlier remark, after the proof of Theorem 8 on p. 525, reads: "It is probably true that $\lim_{k\to\infty}r(C_{2n+1};k)/r(C_3;k)=0$ for $n\ge2$, but this is not known at present." The printed question (v) itself carries no "probably".

The companion question asks whether $r(C_{2n+1};k)$ is at most exponential in $k$ for fixed $n\ge2$; the paper's own bounds (Theorem 7) leave a gap between $2^kn$ and $2(k+2)!\,n$.

Source. P. Erdős and R. L. Graham, On partition theorems for finite graphs, Colloq. Math. Soc. János Bolyai 10 (1975), 515--527; question (v) on printed p. 526 (PDF p. 12 of the archive scan) and the remark on printed p. 525 (PDF p. 11), read on the page images.

Read depth. Claims checked: both passages were read clause by clause on the page images. There is nothing to prove; the questions are posed, not answered, in the paper.

Proof pointer

None; a question. The paper's bounds on the two quantities are Theorems 7 and 8 for the numerator and, for $r(C_3;k)$, nothing beyond the general remark on p. 525 that $e^{c_1kn}<r(K_n;k)<k^{c_2kn}$ (the paper's [1]).

Dependencies

None.

Bears on

• Problem 554: the origin of the problem in the authors' own words, with the weaker companion question; Erdős's 1981 survey restates the conjecture as its item (11).